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Complex number

A complex number is a number of the form $z=x+iy$, where $x$ and $y$ are real numbers (cf.
Real number) and $i=\def\i{\sqrt{-1}}\i$ is the so-called imaginary unit,
that is, a number whose square is equal to $-1$ (in engineering
literature, the notation $j=\i$ is also used): $x$ is called the real
part of the complex number $z$ and $ y$ its imaginary part (written
$x=\def\Re{\mathrm{Re}\;}\Re z$, $y=\def\Im{\mathrm{Im}\;}\Im z$). The real numbers can be regarded as special complex
numbers, namely those with $y=0$. Complex numbers that are not real,
that is, for which $y\ne 0$, are sometimes called imaginary numbers. The
complicated historical process of the development of the notion of a
complex number is reflected in the above terminology which is mainly
of traditional origin.

Algebraically speaking, a complex number is an element of the
(algebraic) extension $\C$ of the field of real numbers $\R$ obtained by
the adjunction to the field $\R$ of a root $i$ of the polynomial
$X^2+1$. The field $\C$ obtained in this way is called the field of complex
numbers or the complex number field. The most important property of
the field $\C$ is that it is algebraically closed, that is, any
polynomial with coefficients in $\C$ splits into linear factors. The
property of being algebraically closed can be expressed in other words
by saying that any polynomial of degree $n\ge 1$ with coefficients in $\C$
has at least one root in $\C$ (the d'Alembert–Gauss theorem or
fundamental theorem of algebra).

The field $\C$ can be constructed as follows. The elements $z=(x,y)$, $z'=(x',y'),\dots$ or
complex numbers, are taken to be the points $z=(x,y)$, $z'=(x',y'),\dots$ of the plane $\R^2$
in Cartesian rectangular coordinates $x$ and $y$, $x'$ and $y',\dots$. Here
the sum of two complex numbers $z=(x,y)$ and $z'=(x',y')$ is the complex number $(x+x',y+y')$,
that is,
$$z+z'=(x,y)+(x',y')=(x+x',y+y'),\label{1}$$
and the product of those complex numbers is the complex
number $(xx'-yy',xy'+x'y)$, that is,
$$zz'=(x,y)(x'y') = (xx'-yy',xy'+x'y).\label{2}$$
The zero element $0=(0,0)$ is the same as the
origin of coordinates, and the complex number $(1,0)$ is the identity of
$\C$.

The plane $\R^2$ whose points are identified with the elements of $\C$ is
called the complex plane. The real numbers $x,x',\dots$ are identified here
with the points $(x,0)$, $(x',0),\dots$ of the $x$-axis which, when referring to the
complex plane, is called the real axis. The points $(0,y)=iy$, $(0,y')=iy',\dots$ are
situated on the $y$-axis, called the imaginary axis of the complex
plane $\C$; numbers of the form $iy,iy',\dots$ are called pure imaginary. The
representation of elements $z,z',\dots$ of $\C$, or complex numbers, as points
of the complex plane with the rules (1) and (2) is equivalent to the
above more widely used form of notating complex numbers:
$$z=(x,y)=x+iy, z'=(x',y')=x'+iy',\dots,$$
also
called the algebraic or Cartesian form of writing complex
numbers. With reference to the algebraic form, the rules (1) and (2)
reduce to the simple condition that all operations with complex
numbers are carried out as for polynomials, taking into account the
property of the imaginary unit: $ii=i^2=-1$.

The complex numbers $z=(x,y)=x+iy$ and $\bar z=(x,-y)=x-iy$ are called conjugate or [[complex
conjugate]]s in the plane $\C$; they are symmetrically situated with
respect to the real axis. The sum and the product of two conjugate
complex numbers are the real numbers
$$z+\bar z = 2\Re z,\quad z\bar z=|z|^2,$$
where $|z|=r=\sqrt{x^2+y^2}$ is called the
modulus or absolute value of $z$.

The following inequalities always hold:
$$|z|-|z'| \le |z+z'|\le |z|+|z'|.$$
A complex number $z$ is
different from 0 if and only if $|z|>0$. The mapping $z\mapsto \bar z$ is an
automorphism of the complex plane of order 2 (that is, $z = \bar{\bar z}$) that
leaves all points of the real axis fixed. Furthermore, $\overline{z+z'} = \bar z + \bar{z'}$, $\bar{zz'} = \bar{z}\bar{z'}$.

The operations of addition and multiplication (1) and (2) are
commutative and associative, they are related by the distributive law,
and they have the inverse operations subtraction and division (except
for division by zero). The latter are expressed in algebraic form as:

$$z-z'=(x+iy)-(x'+iy')=(x-x')+i(y-y'),$$

$$\frac{z'}{z} = \frac{x'+iy'}{x+iy} = \frac{z\bar z}{|z|^2}
=\frac{xx'+yy'}{x^2+y^2}+i\frac{y'x-x'y}{x^2+y^2},\quad z\ne0.\label{3}$$
Division of a complex number $z'$ by a complex number $z\ne0$ thus
reduces to multiplication of $z'$ by
$$\frac{\bar z}{|z|^2} = \frac{x}{x^2+y^2}-i\frac{y}{x^2+y^2}.$$
It is an important question
whether the extension $\C$ of the field of reals constructed above,
with the rules of operation indicated, is the only possible one or
whether essentially different variants are conceivable. The answer is
given by the uniqueness theorem: Every (algebraic) extension of the
field $\R$ obtained from $\R$ by adjoining a root $i$ of the equation
$X^2+1$ is isomorphic to $\C$, that is, only the above rules of operation
with complex numbers are compatible with the requirement that the root
$i$ be algebraically adjoined. This fact, however, does not exclude
the existence of interpretations of complex numbers other than as
points of the complex plane. The following two interpretations are
most frequently employed in applications.

Vector interpretation. A complex number $z=x+iy$ can be identified with the
vector $(x,y)$ with coordinates $x$ and $y$ starting from the origin (see
Fig.).

Figure: c024140a

In this interpretation, addition and subtraction of complex numbers is
carried out according to the rules of addition and subtraction of
vectors. However, multiplication and division of complex numbers,
which must be performed according to (2) and (3), do not have
immediate analogues in vector algebra (see
[Sh],
[LaSc]). The vector interpretation of complex numbers is
immediately applicable, for example, in electrical engineering in the
description of alternating sinusoidal currents and voltages.

Matrix interpretation. The complex number $w=u+iv$ can be identified with a
$(2\times 2)$-matrix of special type
$$w=\begin{pmatrix}\phantom{-}u&v\\ -v&u\end{pmatrix}$$
where the operations of addition,
subtraction and multiplication are carried out according to the usual
rules of matrix algebra.

By using polar coordinates in the complex plane $\C$, that is, the
radius vector $r=|z|$ and polar angle $\def\phi{\varphi}\phi=\arg z$, called here the argument of $z$
(sometimes also called the phase of $z$), one obtains the
trigonometric or polar form of a complex number:
$$z=r(\cos\phi + i\sin\phi),\label{4}$$

$$r\cos\phi = \Re z,\quad r\sin\phi=\Im z.$$
The argument $\phi=\arg z$ is a many-valued real-valued function of the
complex number $z\ne 0z\ne 0$, whose values for a given $z$ differ by integral
multiples of $2\pi i$; the argument of the complex number $z=0$ is not
defined. One usually takes the principal value of the argument $\phi = \def\Arg{\mathrm{Arg}} \Arg z$,
defined by the additional condition $-\pi < \Arg z \le \pi$. The
Euler formulas $e^{\pm i\phi} = \cos\phi\pm i\sin\phi$ transform the trigonometric form
(4) into the exponential form of a complex number:
$$z=re^{i\phi}\label{5}$$
The forms (4)
and (5) are particularly suitable for carrying out multiplication and
division of complex numbers:
$$zz'=rr'[\cos(\phi+\phi')+i\sin(\phi+\phi')]=rr'e^{i(\phi+\phi')},$$

$$\frac{z}{z'}=\frac{r}{r'}[\cos(\phi-\phi')+i\sin(\phi-\phi')]
=\frac{r}{r'}e^{i(\phi-\phi')},\quad r>0$$
Under multiplication (or division) of complex numbers the moduli
are multiplied (or divided) and the arguments are added (or
subtracted). Raising to a power or extracting a root is carried out
according to the so-called de Moivre formulas:
$$z^n = r^n(\cos n\phi + i\sin n\phi) = r^n e^{in\phi},$$

$$k=0,\dots,n-1,$$
where the first of these is also applicable for negative integer
exponents $n$. Geometrically, multiplication of a complex number $z$
by a complex number $z'=r'e^{i\phi'}$ reduces to rotating the vector $z$ over the
angle $\phi'$ (anti-clockwise if $\phi'>0$) and subsequently multiplying its
length by $|z'|=r'$; in particular, multiplication by a complex number $z'=e^{i\phi'}$,
which has modulus one, is merely rotation over the angle $\phi'$. Thus,
complex numbers can be interpreted as operators of a special type
(affinors, cf.
Affinor). In this connection, the mixed vector-matrix
interpretation of multiplication of complex numbers is sometimes
useful:
$$(x,y)\begin{pmatrix}\phantom{-}u&v\\ -v&u\end{pmatrix}=(xu-yv,xv+yu),$$
in which the multiplicand is treated as a matrix-vector
and the multiplier as a matrix-operator.

The bijection $(x,y)\mapsto x+iy$ induces on the field $\C$ the topology of the
$2$-dimensional real vector space $\R^2$; this topology is compatible
with the field structure of $\C$ and thus $\C$ is a topological
field. The modulus $|z|$ is the Euclidean norm of the complex number
$z={x,y}$, and $\C$ endowed with this norm is a complex one-dimensional
Euclidean space, also called the complex $z$-plane. The topological
product $\C^n=\C\times\cdots\times\C$ ($n$ times, $n\ge 1$) is a complex $n$-dimensional Euclidean
space. For a satisfactory analysis of functions it is usually
necessary to consider their behaviour in the complex domain. This is
due to the fact that $\C$ is algebraically closed. Even the behaviour
of such elementary functions as $z^n$, $\cos z$, $\sin z$, $e^z$ can be properly
understood only when they are regarded as functions of a complex
variable (see
Analytic function).

Apparently, imaginary quantities first occurred in the celebrated work
The great art, or the rules of algebra by G. Cardano, 1545, who
regarded them as useless and unsuitable for applications. R. Bombelli
(1572) was the first to realize the value of the use of imaginary
quantities, in particular for the solution of the
cubic equation in the so-called irreducible case
(when the real roots are expressed in terms of cube roots of imaginary
quantities, cf.
Cardano formula). He gave some of the simplest
rules of operation with complex numbers. In general, expressions of
the form $a+b\i$, $b\ne 0$, appearing in the solution of quadratic and cubic
equations were called "imaginary" in the 16th century and 17th
century. However, even for many of the great scholars of the 17th
century, the algebraic and geometric nature of imaginary quantities
was unclear and even mystical. It is known, for example, that
I. Newton did not include imaginary quantities within the notion of
number, and that G. Leibniz said that "complex numbers are a fine and
wonderful refuge of the divine spirit, as if it were an amphibian of
existence and non-existence" .

The problem of expressing the $n$-th roots of a given number was
mainly solved in the papers of A. de Moivre (1707, 1724) and R. Cotes
(1722). The symbol $i=\i$ was proposed by L. Euler (1777, published
1794). It was he who in 1751 asserted that the field $\C$ is
algebraically closed; J. d'Alembert (1747) came to the same
conclusion. The first rigorous proof of this fact is due to C.F. Gauss
(1799), who introduced the term "complex number" in 1831. The complete
geometric interpretation of complex numbers and operations on them
appeared first in the work of C. Wessel (1799). The geometric
representation of complex numbers, sometimes called the "Argand
diagramArgand diagram" , came into use after the publication in 1806
and 1814 of papers by J.R. Argand, who rediscovered, largely
independently, the findings of Wessel.

The purely arithmetic theory of complex numbers as pairs of real
numbers was introduced by W. Hamilton (1837). He found a
generalization of complex numbers, namely the quaternions (cf.
Quaternion), which form a non-commutative algebra. More
generally, it was proved at the end of the 19th century that any
extension of the notion of number beyond the complex numbers requires
sacrificing some property of the usual operations (primarily
commutativity). See also
Hypercomplex number;
Double and dual numbers;
Cayley numbers.